Largest known prime number

The largest known prime number is 2^136,279,841 − 1, a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a 36-year-old researcher from San Jose, California, to the Great Internet Mersenne Prime Search (GIMPS).^[1]^[2]

A prime number is a natural number greater than 1 with no divisors other than 1 and itself. Euclid's theorem proves that for any given prime number, there will always be a higher one, and thus there are infinitely many; there is no largest prime.

Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilize a specialized primality test that is faster than the general one. As of October 2024^[update], the seven largest known primes are Mersenne primes.^[3] The last eighteen record primes were Mersenne primes.^[4]^[5] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2^k − 1 is simply k ones.^[6]

Finding larger prime numbers is sometimes presented as a means to stronger encryption, but this is not the case.^[7]^[8]

^ "GIMPS Project Discovers Largest Known Prime Number: 2^136,279,841-1". Mersenne Research, Inc. 21 October 2024. Retrieved 21 October 2024.
^ Voight, John; Conversation, The. "A 41-million-digit prime number is the biggest ever found—but mathematicians' search for perfection will continue". phys.org. Retrieved 2025-01-14.
^ "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
^ Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
^ The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2^216,193 − 1; see also The Largest Known Prime by year: A Brief History originally by Caldwell.
^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...
^ McKinnon, Mika (January 4, 2018). "This Is the Largest Known Prime Number Yet". Smithsonian. Retrieved July 6, 2024.
^ Johnston, Nathaniel (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography". njohnston.ca. Retrieved July 6, 2024.

[GIMPS-2024-1] "GIMPS Project Discovers Largest Known Prime Number: 2^136,279,841-1". Mersenne Research, Inc. 21 October 2024. Retrieved 21 October 2024.

[2] Voight, John; Conversation, The. "A 41-million-digit prime number is the biggest ever found—but mathematicians' search for perfection will continue". phys.org. Retrieved 2025-01-14.

[3] "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.

[computer_history-4] Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.

[5] The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2^216,193 − 1; see also The Largest Known Prime by year: A Brief History originally by Caldwell.

[6] "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...

[7] McKinnon, Mika (January 4, 2018). "This Is the Largest Known Prime Number Yet". Smithsonian. Retrieved July 6, 2024.

[8] Johnston, Nathaniel (September 11, 2009). "No, Primes with Millions of Digits Are Not Useful for Cryptography". njohnston.ca. Retrieved July 6, 2024.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]